Having been able to compare the different Golden Section spirals
in Part 3, this section looks at the most enduring myth of Golden
Section spirals: because a spiral is a logarithmic spiral it
is a Golden Section one.

"A LITTLE KNOWLEDGE IS A DANGEROUS
THING"Using
the mathematics of the Golden Section spirals in part 3, and
the description of spirals in part 1, I now want to show that
people with a slight knowledge of mathematics can make leaps
of deduction, which can then be perpetuated as myths.

The simplest case of a mistake was one I heard on the radio
recently by an artist who had been brought into one of a series
of programmes on numbers. In the one on the Golden Section he
went on to describe the spiral in the Golden Section rectangle.
This is not an easy thing to do on the radio. It was made harder
that his train of thought was:

The Greeks were the geometers who knew about and discovered
the Golden Section.

Archimedes is a well-known Greek geometer who has a spiral
named after him.

Therefore the spiral is a spiral of Archimedes.

This is quite rare. The most common misconception follows
the logic:

Spirals of shells, particularly the nautilus are logarithmic
spirals.

The Golden Section can be used to draw a logarithmic spiral.

Therefore shell spirals are related to the Golden Section
and in particular, the nautilus shell spiral is a Golden Section
spiral.

I will also describe how a spiral design extrapolated from
the rose window of Chartres cathedral has been misattributed
in the same way.

One of the amazing things about such misconceptions is that
it is so widespread, even by mathematicians who should know better.
It is a prime example of why geometry needs to be taught more
widely and not only geometry, but the visual appreciation of
shape and proportion. This is especially true for the education
of artists, graphic designers and architects.

I can explain one way the errors about the Golden Section
spiral are perpetuated. This anecdote affected me personally.
I wrote an article on Golden Section spirals for a mathematics
magazine a few years ago that included a section on why the nautilus
shell was not a Golden Section spiral. The page proofs came back
with a heading that was quite bland. I was horrified, however,
when I received my sample copy. The designer had drawn a nautilus
shell on the cover and put the same drawing at the heading of
the article. The editorial in the next issue had an interesting
apology and retraction. However, readers do not always see the
next issue.

WHY THE NAUTILUS SHELL IS NOT A
GOLDEN SECTION SPIRALThe
mathematics of the Golden Section spirals in part 3 allows the
spirals to be quantified. We have seen that this has enabled
us to say that each one is different. They are all logarithmic/equiangular
spirals, but the tangent angle a is
different.

The nautilus shell is a logarithmic spiral. Such a shape arises
because a growing animal has the same proportions as it grows
and the spiral fits the requirement to protect this shape as
it gets larger.

When trying to measure a nautilus shell to determine the shape
of the spiral, you can either work from a photograph or a sectioned
specimen, which is essentially equivalent (though photos may
add distortions, of course). In either case, there are numerous
experimental difficulties. For example:

the section may not be exactly in the right direction, that
is it might not be in the right plane;

the thickness of the shell means that there is a considerable
error in deciding where to take the measurement;

finding the centre (pole) of the spiral is not always easy.

The specimen in Figure 37 was measured as follows.

Figure 37

In order to approximate the multiplication factor for the
nautilus logarithmic spiral, measurements were taken for four
different 360° rotations of the spiral and the ratio of the
radial vectors calculated for each rotation. The yellow marks
show the measurement lines. Values obtained were 2.95, 3.02,
2.83, and 2.97, giving an average of 2.94. Although the errors
are quite high it shows a value in the region of 3. Other measurements
I have seen are also around 3. Since the shell is a living form,
making statements other than this is as far as you can go. The
creatures are no more uniformly shaped than you or I are.

A logarithmic spiral with a multiplication factor of 3 has
a tangent angle of 80.08°. Compare this with the following
table:

Golden Section Spiral

tangent angle

multiplication factor

rectangle

72.9676°

f4 » 6.8541

triangle LLS

75.6788°

f10
/ 3 » 4.9731

triangle SSL

79.1609°

f5/2 » 3.3302

pentagon

62.9520°

f20/3 » 24.7315

Note that a small change in tangent angle corresponds to a
large change in the multiplication factor. So although the nautilus
ratio is close to the SSL ratio, this is merely a coincidence.
It is a very long way from the one for the Golden Section rectangle.
In fact, comparing the Golden Section spiral (on the left in
Figure 38 below) with the logarithmic spiral having a multiplication
factor of 3 (on the right in Figure 38 below) and the nautilus
in Figure 37, it is clear that the Golden Section rectangular
spiral and the nautilus spiral simply do not match. There just
are not enough turns with the Golden Section spiral.

Figure 38

Of course, one could specify any multiplication factor and
use it to define a specific logarithmic spiral. In this way,
one could define other Golden Section spirals, without appealing
to any approximate spiral construction, simply by specifying
the multiplication factor to be a chosen power of f
(e.g., f, f2,
f1/2, - the possibilities are
endless). With a suitable choice of the power, one could even
produce a spiral very close to the nautilus spiral. This would
be quite contrived, however, and it must be stressed that the
spiral traditionally associated with the nautilus is the one
corresponding to the Golden Section rectangle (i.e., multiplication
factor f4), which is clearly far from a match.

There is a much longer and more detailed discussion on this
subject in [Fonseca 1993]. He also finds the ratio for one turn
of the nautilus as very close to 3. He describes how an artist
makes errors in order to justify her assumption that the nautilus
is a Golden Section spiral. It is a classic description of how
the Golden Section is misused.

THE NORTH WINDOW OF CHARTRESIn his excellent Rose Windows [Cowen 1979],
Painton Cowen superimposes a geometrical diagram over the north
window of Chartres cathedral and in doing so makes the leap that
since the Golden Section can be used to draw a logarithmic spiral,
this is a Golden Section spiral.

Figure 39

The diagram Cowen uses is shown in Figure 39. Such a set of
spirals bears a superficial resemblance to the sets of pentagon
spirals shown in Figure 29. He has probably made the visual leap
having seen a diagram of the spirals in a sunflower since it
looks so close.

The overall spiral pattern has a twelve-fold symmetry that
matches the symmetry of the window, which reflects the twelve
apostles. He has drawn many auxiliary spirals like the one shown
above, as well. The auxiliary spiral has a multiplication factor
of (Ö2)8 =
16, which leads to a » 66.1895°,
whereas the spirals of the overall pattern have multiplication
factor (Ö2)24
= 4096, which leads to a » 37.0672°.
There is no reason to call either of these spirals a Golden Section
spiral.